/* RISO: an implementation of distributed belief networks.
 * Copyright (C) 1999, Robert Dodier.
 *
 * This program is free software; you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation; either version 2 of the License, or
 * (at your option) any later version.
 *
 * This program is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * GNU General Public License for more details.
 *
 * You should have received a copy of the GNU General Public License
 * along with this program; if not, write to the Free Software
 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA, 02111-1307, USA,
 * or visit the GNU web site, www.gnu.org.
 */
package edu.cmu.lti.algorithm.optimization;

/** <p> This class contains code for the limited-memory Broyden-Fletcher-Goldfarb-Shanno
  * (LBFGS) algorithm for large-scale multidimensional unconstrained minimization problems.
  * This file is a translation of Fortran code written by Jorge Nocedal.
  * The only modification to the algorithm is the addition of a cache to
  * store the result of the most recent line search. See <tt>solution_cache</tt> below.
  *
  * LBFGS is distributed as part of the RISO project. Following is a message from Jorge Nocedal:
  * <pre>
  *   From: Jorge Nocedal [mailto:nocedal@dario.ece.nwu.edu]
  *   Sent: Friday, August 17, 2001 9:09 AM
  *   To: Robert Dodier
  *   Subject: Re: Commercial licensing terms for LBFGS?
  *
  *   Robert:
  *   The code L-BFGS (for unconstrained problems) is in the public domain.
  *   It can be used in any commercial application.
  *
  *   The code L-BFGS-B (for bound constrained problems) belongs to
  *   ACM. You need to contact them for a commercial license. It is
  *   algorithm 778.
  *
  *   Jorge
  * </pre>
  *
  * <p> This code is derived from the Fortran program <code>lbfgs.f</code>.
  * The Java translation was effected mostly mechanically, with some
  * manual clean-up; in particular, array indices start at 0 instead of 1.
  * Most of the comments from the Fortran code have been pasted in here
  * as well.</p>
  *
  * <p> Here's some information on the original LBFGS Fortran source code,
  * available at <a href="http://www.netlib.org/opt/lbfgs_um.shar">
  * http://www.netlib.org/opt/lbfgs_um.shar</a>. This info is taken
  * verbatim from the Netlib blurb on the Fortran source.</p>
  *
  * <pre>
  * 	file    opt/lbfgs_um.shar
  * 	for     unconstrained optimization problems
  * 	alg     limited memory BFGS method
  * 	by      J. Nocedal
  * 	contact nocedal@eecs.nwu.edu
  * 	ref     D. C. Liu and J. Nocedal, ``On the limited memory BFGS method for
  * 	,       large scale optimization methods'' Mathematical Programming 45
  * 	,       (1989), pp. 503-528.
  * 	,       (Postscript file of this paper is available via anonymous ftp
  * 	,       to eecs.nwu.edu in the directory pub/lbfgs/lbfgs_um.)
  * </pre>
  *
  * @author Jorge Nocedal: original Fortran version, including comments
  * (July 1990). Robert Dodier: Java translation, August 1997.
  */

public class LBFGS
{
    /** Specialized exception class for LBFGS; contains the
      * <code>iflag</code> value returned by <code>lbfgs</code>.
      */

    public static class ExceptionWithIflag extends Exception
    {
		private static final long serialVersionUID = 1L;
		public int iflag;
        public ExceptionWithIflag( int i, String s ) { super(s); iflag = i; }
        public String toString() { return getMessage()+" (iflag == "+iflag+")"; }
    }

    /** Controls the accuracy of the line search <code>mcsrch</code>. If the
      * function and gradient evaluations are inexpensive with respect
      * to the cost of the iteration (which is sometimes the case when
      * solving very large problems) it may be advantageous to set <code>gtol</code>
      * to a small value. A typical small value is 0.1.  Restriction:
      * <code>gtol</code> should be greater than 1e-4.
      */

    public static double gtol = 0.9;

    /** Specify lower bound for the step in the line search.
      * The default value is 1e-20. This value need not be modified unless
      * the exponent is too large for the machine being used, or unless
      * the problem is extremely badly scaled (in which case the exponent
      * should be increased).
      */

    public static double stpmin = 1e-20;

    /** Specify upper bound for the step in the line search.
      * The default value is 1e20. This value need not be modified unless
      * the exponent is too large for the machine being used, or unless
      * the problem is extremely badly scaled (in which case the exponent
      * should be increased).
      */

    public static double stpmax = 1e20;

    /** The solution vector as it was at the end of the most recently
      * completed line search. This will usually be different from the
      * return value of the parameter <tt>x</tt> of <tt>lbfgs</tt>, which
      * is modified by line-search steps. A caller which wants to stop the
      * optimization iterations before <tt>LBFGS.lbfgs</tt> automatically stops
      * (by reaching a very small gradient) should copy this vector instead
      * of using <tt>x</tt>. When <tt>LBFGS.lbfgs</tt> automatically stops,
      * then <tt>x</tt> and <tt>solution_cache</tt> are the same.
      */
    public static double[] solution_cache = null;

    private static double gnorm = 0, stp1 = 0, ftol = 0, stp[] = new double[1], ys = 0, yy = 0, sq = 0, yr = 0, beta = 0, xnorm = 0;
    private static int iter = 0, nfun = 0, point = 0, ispt = 0, iypt = 0, maxfev = 0, info[] = new int[1], bound = 0, npt = 0, cp = 0, i = 0, nfev[] = new int[1], inmc = 0, iycn = 0, iscn = 0;
    private static boolean finish = false;

    private static double[] w = null;

    /** This method returns the total number of evaluations of the objective
      * function since the last time LBFGS was restarted. The total number of function
      * evaluations increases by the number of evaluations required for the
      * line search; the total is only increased after a successful line search.
      */
    public static int nfevaluations() { return nfun; }

    /** This subroutine solves the unconstrained minimization problem
      * <pre>
      *     min f(x),    x = (x1,x2,...,x_n),
      * </pre>
      * using the limited-memory BFGS method. The routine is especially
      * effective on problems involving a large number of variables. In
      * a typical iteration of this method an approximation <code>Hk</code> to the
      * inverse of the Hessian is obtained by applying <code>m</code> BFGS updates to
      * a diagonal matrix <code>Hk0</code>, using information from the previous M steps.
      * The user specifies the number <code>m</code>, which determines the amount of
      * storage required by the routine. The user may also provide the
      * diagonal matrices <code>Hk0</code> if not satisfied with the default choice.
      * The algorithm is described in "On the limited memory BFGS method
      * for large scale optimization", by D. Liu and J. Nocedal,
      * Mathematical Programming B 45 (1989) 503-528.
      *
      * The user is required to calculate the function value <code>f</code> and its
      * gradient <code>g</code>. In order to allow the user complete control over
      * these computations, reverse  communication is used. The routine
      * must be called repeatedly under the control of the parameter
      * <code>iflag</code>.
      *
      * The steplength is determined at each iteration by means of the
      * line search routine <code>mcsrch</code>, which is a slight modification of
      * the routine <code>CSRCH</code> written by More' and Thuente.
      *
      * The only variables that are machine-dependent are <code>xtol</code>,
      * <code>stpmin</code> and <code>stpmax</code>.
      *
      * Progress messages and non-fatal error messages are printed to <code>System.err</code>.
      * Fatal errors cause exception to be thrown, as listed below.
      *
      * @param n The number of variables in the minimization problem.
      *		Restriction: <code>n &gt; 0</code>.
      *
      * @param m The number of corrections used in the BFGS update.
      *		Values of <code>m</code> less than 3 are not recommended;
      *		large values of <code>m</code> will result in excessive
      *		computing time. <code>3 &lt;= m &lt;= 7</code> is recommended.
      *		Restriction: <code>m &gt; 0</code>.
      *
      * @param x On initial entry this must be set by the user to the values
      *		of the initial estimate of the solution vector. On exit with
      *		<code>iflag = 0</code>, it contains the values of the variables
      *		at the best point found (usually a solution).
      *
      * @param f Before initial entry and on a re-entry with <code>iflag = 1</code>,
      *		it must be set by the user to contain the value of the function
      *		<code>f</code> at the point <code>x</code>.
      *
      * @param g Before initial entry and on a re-entry with <code>iflag = 1</code>,
      *		it must be set by the user to contain the components of the
      *		gradient <code>g</code> at the point <code>x</code>.
      *
      * @param diagco  Set this to <code>true</code> if the user  wishes to
      *		provide the diagonal matrix <code>Hk0</code> at each iteration.
      *		Otherwise it should be set to <code>false</code> in which case
      *		<code>lbfgs</code> will use a default value described below. If
      *		<code>diagco</code> is set to <code>true</code> the routine will
      *		return at each iteration of the algorithm with <code>iflag = 2</code>,
      *		and the diagonal matrix <code>Hk0</code> must be provided in
      *		the array <code>diag</code>.
      *
      * @param diag If <code>diagco = true</code>, then on initial entry or on
      *		re-entry with <code>iflag = 2</code>, <code>diag</code>
      *		must be set by the user to contain the values of the
      *		diagonal matrix <code>Hk0</code>. Restriction: all elements of
      *		<code>diag</code> must be positive.
      *
      * @param iprint Specifies output generated by <code>lbfgs</code>.
      *		<code>iprint[0]</code> specifies the frequency of the output:
      *		<ul>
      *		<li> <code>iprint[0] &lt; 0</code>: no output is generated,
      *		<li> <code>iprint[0] = 0</code>: output only at first and last iteration,
      *		<li> <code>iprint[0] &gt; 0</code>: output every <code>iprint[0]</code> iterations.
      *		</ul>
      *
      *		<code>iprint[1]</code> specifies the type of output generated:
      *		<ul>
      *		<li> <code>iprint[1] = 0</code>: iteration count, number of function
      *			evaluations, function value, norm of the gradient, and steplength,
      *		<li> <code>iprint[1] = 1</code>: same as <code>iprint[1]=0</code>, plus vector of
      *			variables and  gradient vector at the initial point,
      *		<li> <code>iprint[1] = 2</code>: same as <code>iprint[1]=1</code>, plus vector of
      *			variables,
      *		<li> <code>iprint[1] = 3</code>: same as <code>iprint[1]=2</code>, plus gradient vector.
      *		</ul>
      *
      *	@param eps Determines the accuracy with which the solution
      *		is to be found. The subroutine terminates when
      *		<pre>
      *            ||G|| &lt; EPS max(1,||X||),
      *		</pre>
      *		where <code>||.||</code> denotes the Euclidean norm.
      *
      *	@param xtol An estimate of the machine precision (e.g. 10e-16 on a
      *		SUN station 3/60). The line search routine will terminate if the
      *		relative width of the interval of uncertainty is less than
      *		<code>xtol</code>.
      *
      * @param iflag This must be set to 0 on initial entry to <code>lbfgs</code>.
      *		A return with <code>iflag &lt; 0</code> indicates an error,
      *		and <code>iflag = 0</code> indicates that the routine has
      *		terminated without detecting errors. On a return with
      *		<code>iflag = 1</code>, the user must evaluate the function
      *		<code>f</code> and gradient <code>g</code>. On a return with
      *		<code>iflag = 2</code>, the user must provide the diagonal matrix
      *		<code>Hk0</code>.
      *
      *		The following negative values of <code>iflag</code>, detecting an error,
      *		are possible:
      *		<ul>
      *		<li> <code>iflag = -1</code> The line search routine
      *			<code>mcsrch</code> failed. One of the following messages
      *			is printed:
      *			<ul>
      *			<li> Improper input parameters.
      *			<li> Relative width of the interval of uncertainty is at
      *				most <code>xtol</code>.
      *			<li> More than 20 function evaluations were required at the
      *				present iteration.
      *			<li> The step is too small.
      *			<li> The step is too large.
      *			<li> Rounding errors prevent further progress. There may not
      *				be  a step which satisfies the sufficient decrease and
      *				curvature conditions. Tolerances may be too small.
      *			</ul>
      *		<li><code>iflag = -2</code> The i-th diagonal element of the diagonal inverse
      *			Hessian approximation, given in DIAG, is not positive.
      *		<li><code>iflag = -3</code> Improper input parameters for LBFGS
      *			(<code>n</code> or <code>m</code> are not positive).
      *		</ul>
      *
      *	@throws LBFGS.ExceptionWithIflag
      */

    public static void lbfgs ( int n , int m , double[] x , double f , double[] g , boolean diagco , double[] diag , int[] iprint , double eps , double xtol , int[] iflag ) throws ExceptionWithIflag
    {
        boolean execute_entire_while_loop = false;

        if ( w == null || w.length != n*(2*m+1)+2*m )
        {
            w = new double[ n*(2*m+1)+2*m ];
        }

        if ( iflag[0] == 0 )
        {
            // Initialize.

            solution_cache = new double[n];
            System.arraycopy( x, 0, solution_cache, 0, n );

            iter = 0;

            if ( n <= 0 || m <= 0 )
            {
                iflag[0]= -3;
                throw new ExceptionWithIflag( iflag[0], "Improper input parameters  (n or m are not positive.)" );
            }

            if ( gtol <= 0.0001 )
            {
                System.err.println( "LBFGS.lbfgs: gtol is less than or equal to 0.0001. It has been reset to 0.9." );
                gtol= 0.9;
            }

            nfun= 1;
            point= 0;
            finish= false;

            if ( diagco )
            {
                for ( i = 1 ; i <= n ; i += 1 )
                {
                    if ( diag [ i -1] <= 0 )
                    {
                        iflag[0]=-2;
                        throw new ExceptionWithIflag( iflag[0], "The "+i+"-th diagonal element of the inverse hessian approximation is not positive." );
                    }
                }
            }
            else
            {
                for ( i = 1 ; i <= n ; i += 1 )
                {
                    diag [ i -1] = 1;
                }
            }
            ispt= n+2*m;
            iypt= ispt+n*m;

            for ( i = 1 ; i <= n ; i += 1 )
            {
                w [ ispt + i -1] = - g [ i -1] * diag [ i -1];
            }

            gnorm = Math.sqrt ( ddot ( n , g , 0, 1 , g , 0, 1 ) );
            stp1= 1/gnorm;
            ftol= 0.0001;
            maxfev= 20;

            if ( iprint [ 1 -1] >= 0 ) lb1 ( iprint , iter , nfun , gnorm , n , m , x , f , g , stp , finish );

            execute_entire_while_loop = true;
        }

        while ( true )
        {
            if ( execute_entire_while_loop )
            {
                iter= iter+1;
                info[0]=0;
                bound=iter-1;
                if ( iter != 1 )
                {
                    if ( iter > m ) bound = m;
                    ys = ddot ( n , w , iypt + npt , 1 , w , ispt + npt , 1 );
                    if ( ! diagco )
                    {
                        yy = ddot ( n , w , iypt + npt , 1 , w , iypt + npt , 1 );

                        for ( i = 1 ; i <= n ; i += 1 )
                        {
                            diag [ i -1] = ys / yy;
                        }
                    }
                    else
                    {
                        iflag[0]=2;
                        return;
                    }
                }
            }

            if ( execute_entire_while_loop || iflag[0] == 2 )
            {
                if ( iter != 1 )
                {
                    if ( diagco )
                    {
                        for ( i = 1 ; i <= n ; i += 1 )
                        {
                            if ( diag [ i -1] <= 0 )
                            {
                                iflag[0]=-2;
                                throw new ExceptionWithIflag( iflag[0], "The "+i+"-th diagonal element of the inverse hessian approximation is not positive." );
                            }
                        }
                    }
                    cp= point;
                    if ( point == 0 ) cp = m;
                    w [ n + cp -1] = 1 / ys;

                    for ( i = 1 ; i <= n ; i += 1 )
                    {
                        w [ i -1] = - g [ i -1];
                    }

                    cp= point;

                    for ( i = 1 ; i <= bound ; i += 1 )
                    {
                        cp=cp-1;
                        if ( cp == - 1 ) cp = m - 1;
                        sq = ddot ( n , w , ispt + cp * n , 1 , w , 0 , 1 );
                        inmc=n+m+cp+1;
                        iycn=iypt+cp*n;
                        w [ inmc -1] = w [ n + cp + 1 -1] * sq;
                        daxpy ( n , - w [ inmc -1] , w , iycn , 1 , w , 0 , 1 );
                    }

                    for ( i = 1 ; i <= n ; i += 1 )
                    {
                        w [ i -1] = diag [ i -1] * w [ i -1];
                    }

                    for ( i = 1 ; i <= bound ; i += 1 )
                    {
                        yr = ddot ( n , w , iypt + cp * n , 1 , w , 0 , 1 );
                        beta = w [ n + cp + 1 -1] * yr;
                        inmc=n+m+cp+1;
                        beta = w [ inmc -1] - beta;
                        iscn=ispt+cp*n;
                        daxpy ( n , beta , w , iscn , 1 , w , 0 , 1 );
                        cp=cp+1;
                        if ( cp == m ) cp = 0;
                    }

                    for ( i = 1 ; i <= n ; i += 1 )
                    {
                        w [ ispt + point * n + i -1] = w [ i -1];
                    }
                }

                nfev[0]=0;
                stp[0]=1;
                if ( iter == 1 ) stp[0] = stp1;

                for ( i = 1 ; i <= n ; i += 1 )
                {
                    w [ i -1] = g [ i -1];
                }
            }

            Mcsrch.mcsrch ( n , x , f , g , w , ispt + point * n , stp , ftol , xtol , maxfev , info , nfev , diag );

            if ( info[0] == - 1 )
            {
                iflag[0]=1;
                return;
            }

            if ( info[0] != 1 )
            {
                iflag[0]=-1;
                throw new ExceptionWithIflag( iflag[0], "Line search failed. See documentation of routine mcsrch. Error return of line search: info = "+info[0]+" Possible causes: function or gradient are incorrect, or incorrect tolerances." );
            }

            nfun= nfun + nfev[0];
            npt=point*n;

            for ( i = 1 ; i <= n ; i += 1 )
            {
                w [ ispt + npt + i -1] = stp[0] * w [ ispt + npt + i -1];
                w [ iypt + npt + i -1] = g [ i -1] - w [ i -1];
            }

            point=point+1;
            if ( point == m ) point = 0;

            gnorm = Math.sqrt ( ddot ( n , g , 0 , 1 , g , 0 , 1 ) );
            xnorm = Math.sqrt ( ddot ( n , x , 0 , 1 , x , 0 , 1 ) );
            xnorm = Math.max ( 1.0 , xnorm );

            if ( gnorm / xnorm <= eps ) finish = true;

            if ( iprint [ 1 -1] >= 0 ) lb1 ( iprint , iter , nfun , gnorm , n , m , x , f , g , stp , finish );

            // Cache the current solution vector. Due to the spaghetti-like
            // nature of this code, it's not possible to quit here and return;
            // we need to go back to the top of the loop, and eventually call
            // mcsrch one more time -- but that will modify the solution vector.
            // So we need to keep a copy of the solution vector as it was at
            // the completion (info[0]==1) of the most recent line search.

            System.arraycopy( x, 0, solution_cache, 0, n );

            if ( finish )
            {
                iflag[0]=0;
                    return;
            }

            execute_entire_while_loop = true;		// from now on, execute whole loop
        }
    }

    /** Print debugging and status messages for <code>lbfgs</code>.
      * Depending on the parameter <code>iprint</code>, this can include
      * number of function evaluations, current function value, etc.
      * The messages are output to <code>System.err</code>.
      *
      * @param iprint Specifies output generated by <code>lbfgs</code>.<p>
      *		<code>iprint[0]</code> specifies the frequency of the output:
      *		<ul>
      *		<li> <code>iprint[0] &lt; 0</code>: no output is generated,
      *		<li> <code>iprint[0] = 0</code>: output only at first and last iteration,
      *		<li> <code>iprint[0] &gt; 0</code>: output every <code>iprint[0]</code> iterations.
      *		</ul><p>
      *
      *		<code>iprint[1]</code> specifies the type of output generated:
      *		<ul>
      *		<li> <code>iprint[1] = 0</code>: iteration count, number of function
      *			evaluations, function value, norm of the gradient, and steplength,
      *		<li> <code>iprint[1] = 1</code>: same as <code>iprint[1]=0</code>, plus vector of
      *			variables and  gradient vector at the initial point,
      *		<li> <code>iprint[1] = 2</code>: same as <code>iprint[1]=1</code>, plus vector of
      *			variables,
      *		<li> <code>iprint[1] = 3</code>: same as <code>iprint[1]=2</code>, plus gradient vector.
      *		</ul>
      * @param iter Number of iterations so far.
      * @param nfun Number of function evaluations so far.
      * @param gnorm Norm of gradient at current solution <code>x</code>.
      * @param n Number of free parameters.
      * @param m Number of corrections kept.
      * @param x Current solution.
      * @param f Function value at current solution.
      * @param g Gradient at current solution <code>x</code>.
      * @param stp Current stepsize.
      * @param finish Whether this method should print the ``we're done'' message.
      */
    public static void lb1 ( int[] iprint , int iter , int nfun , double gnorm , int n , int m , double[] x , double f , double[] g , double[] stp , boolean finish )
    {
        int i;

        if ( iter == 0 )
        {
            System.err.println( "*************************************************" );
            System.err.println( "  n = " + n + "   number of corrections = " + m + "\n       initial values" );
            System.err.println( " f =  " + f + "   gnorm =  " + gnorm );
            if ( iprint [ 2 -1] >= 1 )
            {
                System.err.print( " vector x =" );
                for ( i = 1; i <= n; i++ )
                    System.err.print( "  "+x[i-1] );
                System.err.println( "" );

                System.err.print( " gradient vector g =" );
                for ( i = 1; i <= n; i++ )
                    System.err.print( "  "+g[i-1] );
                System.err.println( "" );
            }
            System.err.println( "*************************************************" );
            System.err.println( "\ti\tnfn\tfunc\tgnorm\tsteplength" );
        }
        else
        {
            if ( ( iprint [ 1 -1] == 0 ) && ( iter != 1 && ! finish ) ) return;
            if ( iprint [ 1 -1] != 0 )
            {
                if ( (iter - 1) % iprint [ 1 -1] == 0 || finish )
                {
                    if ( iprint [ 2 -1] > 1 && iter > 1 )
                        System.err.println( "\ti\tnfn\tfunc\tgnorm\tsteplength" );
                    System.err.println( "\t"+iter+"\t"+nfun+"\t"+f+"\t"+gnorm+"\t"+stp[0] );
                }
                else
                {
                    return;
                }
            }
            else
            {
                if ( iprint [ 2 -1] > 1 && finish )
                        System.err.println( "\ti\tnfn\tfunc\tgnorm\tsteplength" );
                System.err.println( "\t"+iter+"\t"+nfun+"\t"+f+"\t"+gnorm+"\t"+stp[0] );
            }
            if ( iprint [ 2 -1] == 2 || iprint [ 2 -1] == 3 )
            {
                if ( finish )
                {
                    System.err.print( " final point x =" );
                }
                else
                {
                    System.err.print( " vector x =  " );
                }
                for ( i = 1; i <= n; i++ )
                    System.err.print( "  "+x[i-1] );
                System.err.println( "" );
                if ( iprint [ 2 -1] == 3 )
                {
                    System.err.print( " gradient vector g =" );
                    for ( i = 1; i <= n; i++ )
                        System.err.print( "  "+g[i-1] );
                    System.err.println( "" );
                }
            }
            if ( finish )
                System.err.println( " The minimization terminated without detecting errors. iflag = 0" );
        }
        return;
    }

    /** Compute the sum of a vector times a scalara plus another vector.
      * Adapted from the subroutine <code>daxpy</code> in <code>lbfgs.f</code>.
      * There could well be faster ways to carry out this operation; this
      * code is a straight translation from the Fortran.
      */
    public static void daxpy ( int n , double da , double[] dx , int ix0, int incx , double[] dy , int iy0, int incy )
    {
        int i, ix, iy, m, mp1;

        if ( n <= 0 ) return;

        if ( da == 0 ) return;

        if  ( ! ( incx == 1 && incy == 1 ) )
        {
            ix = 1;
            iy = 1;

            if ( incx < 0 ) ix = ( - n + 1 ) * incx + 1;
            if ( incy < 0 ) iy = ( - n + 1 ) * incy + 1;

            for ( i = 1 ; i <= n ; i += 1 )
            {
                dy [ iy0+iy -1] = dy [ iy0+iy -1] + da * dx [ ix0+ix -1];
                ix = ix + incx;
                iy = iy + incy;
            }

            return;
        }

        m = n % 4;
        if ( m != 0 )
        {
            for ( i = 1 ; i <= m ; i += 1 )
            {
                dy [ iy0+i -1] = dy [ iy0+i -1] + da * dx [ ix0+i -1];
            }

            if ( n < 4 ) return;
        }

        mp1 = m + 1;
        for ( i = mp1 ; i <= n ; i += 4 )
        {
            dy [ iy0+i -1] = dy [ iy0+i -1] + da * dx [ ix0+i -1];
            dy [ iy0+i + 1 -1] = dy [ iy0+i + 1 -1] + da * dx [ ix0+i + 1 -1];
            dy [ iy0+i + 2 -1] = dy [ iy0+i + 2 -1] + da * dx [ ix0+i + 2 -1];
            dy [ iy0+i + 3 -1] = dy [ iy0+i + 3 -1] + da * dx [ ix0+i + 3 -1];
        }
        return;
    }

    /** Compute the dot product of two vectors.
      * Adapted from the subroutine <code>ddot</code> in <code>lbfgs.f</code>.
      * There could well be faster ways to carry out this operation; this
      * code is a straight translation from the Fortran.
      */
    public static double ddot ( int n, double[] dx, int ix0, int incx, double[] dy, int iy0, int incy )
    {
        double dtemp;
        int i, ix, iy, m, mp1;

        dtemp = 0;

        if ( n <= 0 ) return 0;

        if ( !( incx == 1 && incy == 1 ) )
        {
            ix = 1;
            iy = 1;
            if ( incx < 0 ) ix = ( - n + 1 ) * incx + 1;
            if ( incy < 0 ) iy = ( - n + 1 ) * incy + 1;
            for ( i = 1 ; i <= n ; i += 1 )
            {
                dtemp = dtemp + dx [ ix0+ix -1] * dy [ iy0+iy -1];
                ix = ix + incx;
                iy = iy + incy;
            }
            return dtemp;
        }

        m = n % 5;
        if ( m != 0 )
        {
            for ( i = 1 ; i <= m ; i += 1 )
            {
                dtemp = dtemp + dx [ ix0+i -1] * dy [ iy0+i -1];
            }
            if ( n < 5 ) return dtemp;
        }

        mp1 = m + 1;
        for ( i = mp1 ; i <= n ; i += 5 )
        {
            dtemp = dtemp + dx [ ix0+i -1] * dy [ iy0+i -1] + dx [ ix0+i + 1 -1] * dy [ iy0+i + 1 -1] + dx [ ix0+i + 2 -1] * dy [ iy0+i + 2 -1] + dx [ ix0+i + 3 -1] * dy [ iy0+i + 3 -1] + dx [ ix0+i + 4 -1] * dy [ iy0+i + 4 -1];
        }

        return dtemp;
    }
}
